Domain, Range, Maximum and Minimum Radical Functions Domain, Range, Maximum and Minimum

What is a radical function? Definition: Any expression of the form 𝑛 𝑎 denoting the principal nth root of 𝑎 where 𝑛>1. 𝑛 𝑎 ;𝑤ℎ𝑒𝑟𝑒 𝑛=2, 3, 4, 5, … What is the most common form of the equation? 𝑓 𝑥 = 𝑥 = 𝑥 1 2

Square Root of X: 𝑓 𝑥 = 𝑥 Domain: [0,∞) Range: [0,∞) 𝑦= 𝑥 X Y -1 ----- 𝑦= 𝑥 X Y -1 ----- 1 4 2 9 3 16 25 5 Domain: [0,∞) Range: [0,∞)

Cube Root of X: 𝑓 𝑥 = 3 𝑥 Domain: (−∞,∞) Range: (−∞,∞) X Y -27 -3 -8 -1 1 8 2 27 3 Domain: (−∞,∞) Range: (−∞,∞)

Other Radical Functions 𝑓 𝑥 = 𝑥 𝑓 𝑥 = 3 𝑥 𝑓 𝑥 = 4 𝑥

Looking at the Radical 𝑛 𝑥 ;𝑤ℎ𝑒𝑟𝑒 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑛 𝑥 ;𝑤ℎ𝑒𝑟𝑒 𝑛 𝑖𝑠 𝑜𝑑𝑑 0 =0 n is even – 2, 4, 6, … Restrictions: What can’t the square root be? Negative!!! Can it be 0? ( 0 ) Yes!!! 0 =0 n is odd – 3, 5, 7, … Restrictions? Can x be negative? Yes!!! Can it be 0? ( 0 ) 0 =0

Domain and Range Domain: x-values Range: y-values Lowest x value to greatest x value. Even: 𝑥≥0 x is nonnegative Odd: x can be any real number. Lowest y value to greatest y value. Even: 𝑓 𝑥 ≥0 or 𝑦≥0 Odd: 𝑓(𝑥) or 𝑦 can be any real number.

Absolute Maximum and Minimum Domain: [0,∞) Range:[0,∞) Max: (0, -2) Min: (-3,0)

Finding Domain: 𝑛 𝑥 ;where n is even Review – Restrictions: What can’t the square root be? Negative!!! So, the value under the square root must be ≥0. Example: 𝑓 𝑥 = 𝑥−5 𝑥−5≥0→𝒙≥𝟓 Domain: [𝟓,∞) Range: [𝟎,∞) 𝑓 𝑥 = 3𝑥+6 3𝑥+6≥0→3𝑥≥−6→𝒙≥−𝟐 Domain: [−𝟐,∞)

Multiplying or Dividing by a Negative More Examples: Multiplying or Dividing by a Negative Graphs 𝑓 𝑥 = 4−𝑥 4−𝑥≥0→−𝑥≥−4→𝒙≤𝟒 Domain: (−∞,4] Range: [0,∞) 𝑓 𝑥 = − 1 2 𝑥−7 − 1 2 𝑥−7≥0→− 1 2 𝑥≥7 →𝒙≤−𝟏𝟒 Domain: (−∞,−14]

More Examples: 𝑓 𝑥 = 3 𝑥−1 𝑓 𝑥 = 4 4𝑥+12 𝑥−1≥0→𝒙≥𝟏 4𝑥+12≥0→4𝑥≥−12 Different Roots: 𝟑 𝒙 , 𝟒 𝒙 , … Graphs 𝑓 𝑥 = 3 𝑥−1 𝑥−1≥0→𝒙≥𝟏 Domain: (−∞,∞) Range: (−∞,∞) 𝑓 𝑥 = 4 4𝑥+12 4𝑥+12≥0→4𝑥≥−12 →𝒙≥−𝟑 Domain: [−3,∞) Range: [0,∞)

Equations where the range change! More Examples: Equations where the range change! Graphs 𝑓 𝑥 = 𝑥+2 −5 𝑥+2≥0→𝒙≥−𝟐 Domain: [−2,∞) Range: [−5,∞) 𝑓 𝑥 = 3−𝑥 +7 3−𝑥≥0→−𝑥≥−3→𝒙≤𝟑 Domain: (−∞,3] Range: [7,∞)